On the converse of the transitivity of modularity
Y. K. Wong · Bulletin of the American Mathematical Society · 1940
Moore's theorem on the transitivity of modularity is as follows: Consider the basis 1 31, $, e; if a positive hermitian matrix e 0 is modular as to e €, then every vector which is modular as to € 0 is modular 2 as to e (that is, Wfl €0 c 2)? € ).In his doctoral thesis, the author establishes the converse of the preceding theorem as a consequence of the Hellinger-Toeplitz theorem.3 In this note, we give a new proof for the converse of the transitivity of modularity, and then deduce the generalized Hellinger-Toeplitz theorem as a corollary.The converse of the transitivity of modularity is, therefore, equivalent to the Hellinger-Toeplitz theorem.We also establish the converse of the transitivity of modularity for matrices, and a theorem on the transitivity of accordance and finiteness.